The channel capacity C, in bits per second (bps), of a wireless channel with bandwidth B in Hz can be represented [1] byC=B log2(1+SNR)  (1)where SNR is the Signal to Noise Ratio experienced by the Receiver.
Capacity boosting by increasing power is unattractive since the capacity improves only logarithmically with the signal to noise ratio. Practically there exist many hardware impairments in the transceiver that create a signal to noise ratio floor which cannot be further improved by increasing the transmitter power. The impairments include oscillator phase noise, timing jitter, non-linearity, residue carrier and sampling frequency offsets between the transmitter and the receiver and the imbalance between the In-phase and Quadrature (I/Q) channels.
Another way to increase the channel capacity, according to (1), is to increase the signal bandwidth B. However, the spectrum is also a precious resource which has to be wisely shared among many users.
Simple modulation schemes such as Binary Phase Shift Keying (BPSK) is power efficient, but spectrum inefficient. To increase the spectrum efficiency, multi-level schemes such as QAM are more desirable. Orthogonal frequency division multiplexing (OFDM) is another spectrum efficient technique. OFDM divides the total signal spectrum into multiple sub-carriers (or tones) without any guard bands in between. Each sub-carrier is independently modulated so that their spectra overlap but the modulated sub-carrier signals are still orthogonal to each other [2].
In practice, the receiver has to deal with the Inter-Symbol Interference (ISI) caused by multi-path propagation, and filtering in the transmitter and receiver to extract the signal data, a process conventionally called ‘equalization’. With the inclusion of a Cyclic Prefix (CP), OFDM transforms an ISI channel into an ISI-free channel in the frequency-domain, significantly simplifying the equalization complexity. OFDM's insensitivity to small timing-offset reduces the need for over-sampling and fine timing tracking.
Bit Interleaved Coded Modulation (BICM) is an attractive compromise between power and spectrum efficiency and decoder complexity [3]. BICM performs similarly to optimal Trellis Coded Modulation (TCM) with simpler decoder complexity. The decoupling of the modulation and channel coding in a BICM scheme also allows the flexibility to select from a wider class of binary error-control codes and more flexibility in coding rate adaptation. For the above reasons, the BICM scheme with QAM OFDM is adopted in many of the current standards and specifications (e.g. [4]).
It is established that soft-decision decoding (SDD) outperforms the hard-decision decoding (HDD). The SDD requires bit log-likelihood ratio (LLR) to be calculated from the received QAM signal. Optimal bit LLR calculation of QAM signals, has been addressed by [5]. To avoid the complexity of optimal LLR calculation, several authors have proposed the approximate LLR calculation based on the Max Log concept [6], [7], [8] and [9].
In gigabit radio systems, such as the specification defined in [4], 5.992×109 or more bit-LLRs have to be calculated in each second. Minimizing the complexity of these calculations is essential in a low-power and low-cost device. In practice, some processing has to be performed by the receiver before LLR can be calculated. In single-carrier (SC) receivers this processing may involve equalization. In OFDM receivers this processing may involve fast Fourier transform (FFT). For both SC and OFDM receivers, the processing creates simple channel models. Let sn(i) be the transmitted value at subcarrier n of OFDM symbol i. The corresponding value zn(i) at the FFT output can be represented byzn(i)=sn(i)hn(i)+ηn(i),where hn(i) is the channel gain and ηn(i) is the noise and interference component. The above simple channel model equally applies to SC receivers by treating SC modulation as a special OFDM with only one subcarrier such that n=1.
No account has been given to the impact of channel estimation (CE) error [10]. The approximate LLRs proposed by Tosato and Bisaglia [11] were identified as the simplest class of high-performance algorithms currently known. The methods in this invention are much simpler than any of the prior arts. Additionally the invention addresses the issues in soft de-mapping rectangular QAM signals where the signal strength on I and Q channels are different, while the prior arts primarily apply to square QAM signals where the I and Q signal strengths are equal.